2. We have a number of different way
of finding the roots if a quadratic
equations
#1. Making a table
#2. Factoring
#3. Completing the Square
Now a new way that comes from completing
the square.
The Quadratic Formula
3. The Quadratic Formula
Solve for x by completing the square.
ax 2 + bx + c = 0
ax 2 + bx + = −c +
b
−c
2
x + x +=
+
a
a
2
b
−c b
b
x + x+ =
+
a
a 2a
2a
2
2
4. The Quadratic Formula
Solve for x by completing the square.
ax 2 + bx + c = 0
ax 2 + bx + = −c +
b
−c
x2 + x + =
+
a
a
2
b
−c b
b
x2 + x + =
+
a
a 2a
2a
b
b2
b 2 − 4ac
x2 + x + 2 =
a
4a
4a 2
2
2
b b 2 − 4ac
x+ =
2a
4a 2
5. The Quadratic Formula
Solve for x by completing the square.
b
b2
b 2 − 4ac
x + x+ 2 =
a
4a
4a 2
2
2
b b 2 − 4ac
x+ =
2a
4a 2
b
b 2 − 4ac
x+
=±
2a
4a 2
−b
b 2 − 4ac
x=
±
2a
2a
− b ± b 2 − 4ac
x=
2a
7. How does it work
Equation:
3x 2 + 5 x + 1 = 0
a=3
b=5
c =1
− b ± b 2 − 4ac
x=
2a
8. How does it work
Equation:
3x + 5 x + 1 = 0
a=3
b=5
c =1
2
− b ± b 2 − 4ac
x=
2a
x=
− ( 5) ±
( 5) 2 − 4( 3)(1)
2( 3)
− 5 ± 25 − 12
x=
6
x=
− 5 ± 13 − 5
13
=
±
6
6
6
9. The Discriminant
The number in the square root of the
quadratic formula.
b − 4ac
2
Given x − 5 x + 6 = 0
2
( − 5)
− 4(1)( 6 )
25 − 24 = 1
2
10. The Discriminant
b − 4ac
2
The Discriminant can be negative, positive or zero
If the Discriminant is positive then there are:
2 real answers.
If the square root is not a prefect square
( for example 25 ),
then there will be 2 irrational roots
( for example 2 ± 5 ).
11. The Discriminant
b 2 − 4ac
The Discriminant can be negative, positive or zero
If the Discriminant is positive,
there are 2 real answers.
If the Discriminant is zero,
there is 1 real answer.
If the Discriminant is negative,
there are 2 complex answers.
complex answer have i.
12. Let’s put all of that
b
information in a chart.
Value of Discriminant
Type and
Number of Roots
D > 0,
D is a perfect square
2 real,
rational roots
(ex: x= 2 and x= -4)
D > 0,
D NOT a perfect square
2 real,
Irrational roots
(x = √13 x= -√13)
D=0
1 real, rational root
(double root)
(ex: x = 5)
D<0
2 complex roots
(complex conjugates)
(x = 2 ± 3i )
2
− 4ac
Sample Graph
of Related Function
14. Describe the roots
Tell me the Discriminant and the type of roots
x2 + 6x + 9 = 0
0, One rational root
15. Describe the roots
Tell me the Discriminant and the type of roots
x2 + 6x + 9 = 0
0, One rational root
x + 3x + 5 = 0
2
-11, Two complex roots
x + 8x − 4 = 0
2
80, Two irrational roots
16. Describe the roots
Tell me the Discriminant and the type of roots
x2 + 6x + 9 = 0
0, One rational root
x + 3x + 5 = 0
2
17. Describe the roots
Tell me the Discriminant and the type of roots
x2 + 6x + 9 = 0
0, One rational root
x + 3x + 5 = 0
2
-11, Two complex roots
18. Describe the roots
Tell me the Discriminant and the type of roots
x2 + 6x + 9 = 0
0, One rational root
x + 3x + 5 = 0
2
-11, Two complex roots
x + 8x − 4 = 0
2
